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This paper presents a pure complementary energy variational method forsolving anti-plane shear problem in finite elasticity. Based on the canonicalduality-triality theory developed by the author, the nonlinear/nonconex partialdifferential equation for the large deformation problem is converted into analgebraic equation in dual space, which can, in principle, be solved to obtaina complete set of stress solutions. Therefore, a general analytical solutionform of the deformation is obtained subjected to a compatibility condition.Applications are illustrated by examples with both convex and nonconvex storedstrain energies governed by quadratic-exponential and power-law materialmodels, respectively. Results show that the nonconvex variational problem couldhave multiple solutions at each material point, the complementary gap functionand the triality theory can be used to identify both global and local extremalsolutions, while the popular (poly-, quasi-, and rank-one) convexities provideonly local minimal criteria, the Legendre-Hadamard condition does not guaranteeuniqueness of solutions. This paper demonstrates again that the purecomplementary energy principle and the triality theory play important roles infinite deformation theory and nonconvex analysis.

Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity